Very weak global solutions to a parabolic–parabolic chemotaxis-system with logistic source

In this paper we study this chemotaxis-system{ut=Δu−χ∇⋅(u∇v)+g(u),x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0, defined in a convex smooth and bounded domain Ω of RnRn, with n≥1n≥1, and endowed with homogeneous Neumann boundary conditions, being χ>0χ>0 and g   a sort of logistic function obeying growth technical restrictions. Specifically, once an appropriate definition of very weak solution is given, our main result deals with the global existence of such solutions for any nonnegative initial data (u0,v0)∈C0(Ω¯)×C2(Ω¯), and under zero-flux boundary condition on v0v0.